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Volume 12 (2016) 


PROGRESS IN PHYSICS 


Issue 1 (January) 


On an Apparent Resolution of the Catt Question 

Stephen J. Crothers 

Tasmania, Australia. E-mail: steve@plasmaresources.com 
Over a number of years there have been some attempts to answer the Catt Question 
within the context of classical electromagnetic theory. None of the authors of these 
attempts agree on the answer to the Catt Question, even though they all invoke the very 
same theory. An attempt at answering the Catt Question appeared in the journal Physics 
Education in 2013, penned by M. Pieraccini and S. Selleri, as a mathematical rendition 
of their earlier non-mathematical version published in IEEE Antennas and Propagation 
Magazine, 2012. The explanation by these two Authors contains violations of classical 
electromagnetic theory, although they claim to have satisfactorily answered the Catt 
Question by means of classical electromagnetic theory. The arguments adduced by 
Pieraccini and Selleri are therefore invalid. 


1 Introduction 

In their article [1] “An apparent paradox: Catt’s anomaly”, the 
Italian authors Pieraccini and Selleri* refer to the Catt Ques- 
tion as “ Catt’s Anomaly’. Their earlier paper is titled ‘Catt’s 
Anomaly’ [2], Although until 2001 “The Catt Question” was 
called “The Catt Anomaly”, it was in fact always a question, 
to be answered. 

The Catt Question [3] pertains to the propagation of a 
Transverse Electromagnitic (TEM) wave along a transmis- 
sion line. Upon closure of a switch, the TEM wave (step) 
travels at the speed of light between the conducting wires 
of the transmission line, from battery to load, as depicted in 
Fig. 1. 

An electric field E appears between the conductors, di- 
rected from the top wire to the bottom wire. This electric field 
is orthogonal to the two parallel wires and moves towards the 
load; thus there are positive charges on the top conductor and 
negative charges on the bottom conductor in the region of the 
transverse electric field. The Catt Question is: Where does 
this new charge come from? [3], 

2 Electron current 

According to classical electromagnetic theory and circuit the- 
ory, electric current in metallic wires is the flow of electrons 
in the wires (conductors), and a magnetic field is generated 
around the conducting wires according to the Right-Hand 
Rule. Since the TEM step travels at the speed of light to- 
wards the load, how does the current in the conducting wires 
keep pace with the TEM wave, if electrons cannot travel at the 
speed of light? The Authors [1] give the following answer, 
“The key idea of the explanation of this apparent 
paradox is related to the great number of elec- 
trons in metal. Although each single electron is 

"Massimiliano Pieraccini, Associate Professor, Department of Electron- 
ics and Telecommunications, University of Florence; Stefano Selleri, Assis- 
tant Professor, University of Florence. 



Fig. 1: An electric field points directly from the top conductor to 
the bottom conductor (from positive charge to negative charge). It is 
therefore orthogonal to the top and bottom parallel conductors. The 
transverse electric field travels from battery to load at the speed of 
light, subject to the dielectric medium between the wires. 

not able to travel at the speed of light, a great 
number of slow electrons are able to produce a 
current as fast as an electromagnetic wave trav- 
elling at the speed of light in the conductor.” 

What do they mean by “current”? They say here that elec- 
trons “ produce a current”. However, the Authors actually as- 
sume the classical electron flow along wires as the meaning 
of electric current in wires, and claim that this current travels 
along the conductors at the speed of light even though the drift 
speed of electrons in the wires is a snail’s pace (e.g. 2mm/s in 
1.0mm copper wire [1]). Strangely, the flow of electrons, al- 
though very slow, produces an electron current that is “as fast 
as an electromagnetic wave travelling at the speed of light 
in the conductor” [1]: after all, a current of electrons is an 
electron current. This impossible duality occurs, they say, be- 
cause the free electron density in the conductors is very high, 
and they derive an equation for electron drift “velocity”. 

Electron drift velocity in a wire is proportional to the vec- 
tor electric field E„, in the wire, which supposedly causes the 
electron drift, 

v = ~pE w (0) 

and so the electron drift velocity and the electric field in the 
wire are collinear but point in opposite directions. The con- 
stant of proportionality p is called the mobility. 


68 


S. Crothers. On an Apparent Resolution of the Catt Question 


Issue 1 (January) 


PROGRESS IN PHYSICS 


Volume 12 (2016) 


The Authors begin with the following equation for elec- 
tron current, 

/ = na 2 vqN , (1) 

where a is the radius of the conductors, v “ the drift velocity 
of the charges (in practice electrons, and the speed is much 
lower than the speed of light)” [1]*, q the elementary charge, 
and N the free electron density in the conductors. 

Since the current, they say, travels at the speed of light, in 
time At = Ax/c they obtain a passage of charge A Q along the 
top conductor, given by. 


make the transverse electric field the battery EMF to drive 
electrons along the wires; at equation (4). 

Then they introduce the “skin effect” [1]: 

“Up to this point, the current has been consid- 
ered constant in the wire section, but in reality 
the current flow tends to be bound to the portion 
of the conductor closer to the surface.” 

The equation for current in the wires they then give as, 

I = 2na6vqN, (lb) 


AQ = I At = I — , (2) 

c 

where Ax is the distance travelled by the TEM step in time At. 
This charge AQ the Authors call “an imbalance of charge” [1] 
because they say it is confined to a leading volume element of 
length Ax in the top conducting wire, and induces equal but 
opposite polarity charge on the bottom conducting wire. 

Using a cylindrical Gaussian surface they next apply 
Gauss’ Law to calculate the magnitude E of the electric field 
E due to AQ in the top conductor, 

A Q 

— =(2naAx)E, (3) 

63 


where eo is the permitivity of free space. Substituting AQ 
from equation (2) and / from equation (1) the Authors obtain, 


2 cepE 
qNa 


(4) 


From equation (4) they conclude, 

“The notable point of this result is that the nec- 
essary speed decreases with the number of elec- 
trons per volume unit N. Therefore, a great num- 
ber of slow electrons are able to generate enough 
unbalanced charge to follow an electromagnetic 
wave travelling at much higher speed.” 

Thus electrons flow slowly in the conducting wires but the 
electron current in the wires is nevertheless flowing along the 
conductors at the speed of light. 

Although equation (4) follows from equations (1), (2) and 
(3) by purely mathematical operations, the transverse electric 
field E cannot drive electrons along the inside or outside of 
the wires. Equations (1), (2) and (4) imply flow of electrons 
along the wires, but the transverse electric field at equation 
(3) is orthogonal to the parallel axes of the top and bottom 
wires. According to classical electrodynamics, free electrons 
in a metallic conductor flow in the direction opposite to the 
direction of the electric field, according to equation (0), not 
orthogonal to the electric field (E + E„,). The Authors con- 
found battery EMF* with the transverse electric field, and so 


’The Authors confound velocity with speed; the latter denoted by |v| = v 
^What EMF is, is another question. 


where 6 is the skin depth, which is frequency dependent. With 
the “skin effect” they still argue that electrons flowing along 
the wire is electric current, orthogonal to the electric field they 
calculated at equation (3), and continue to make that trans- 
verse electric field the driver of the electrons in the conduct- 
ing wires. Using equations (lb), (2) and (3) they then obtain 
the electron drift speed, 


2ceoE 

qN6 


(5) 


although the 2 in the numerator should not in fact appear. 


3 Conclusion 

Pieraccini and Selleri have not answered the Catt Question. 
On the one hand they treat current in the conducting wires as 
electron current but on the other hand they invoke the trans- 
verse electric field between the conducting wires to drive this 
electron current at the electron drift speed. Their analysis vio- 
lates the classical electromagnetic theory they use in their at- 
tempt to prove that what they call “Catt’s Anomaly” is merely 
an “apparent paradox” [1], The real paradox here is their 
claim that very slowly flowing electrons in the wires of a 
transmission line produce an electron current in those wires 
that travels at the speed of light, driven by an elecric field 
orthogonal to those wires. 

“If I have promised to deliver one dozen eggs to 
Oxford, one hour from now, Oxford being 100 
miles away, there is no point in despatching ten 
dozen eggs in a vehicle which travels at only ten 
miles/h” [4], 


Submitted on December 13, 2015 / Accepted on December 14, 2015 


References 

1. Pieraccini M. and Selleri S. An apparent paradox: Catt’s anomaly. 
Physics Education, 2013, v.48 (6), 718-722. 

2. Pieraccini M. and Selleri S. Catt’s anomaly. IEEE Antennas Propag. 
Mag., 2012, v. 54, no. 6, 242-244. 

3. Catt I. The Catt Question, http://www.ivorcatt.co.uk/x54cl.htm 

4. Catt I. The death of electric current. Wireless World, December 1982, 
79-81. Accessed from http://www.ivorcatt.co.uk/xl8jl00.pdf 


S. Crothers. On an Apparent Resolution of the Catt Question 


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